Middle English melten, from Old English meltan (intransitive) "become liquid through heat" (class III strong verb; past tense mealt, past participle molten), from Proto-Germanic *meltanan; fused with Old English gemæltan (Anglian), gemyltan (West Saxon) "make liquid, reduce from a solid to a fluid state by means of heat" (transitive), from Proto-Germanic *gamaltijan (source also of Old Norse melta "to digest").
Both Germanic words are from PIE *meldh- (source also of Sanskrit mrduh "soft, mild," Greek meldein "to melt, make liquid," Latin mollis "soft, mild"), from root *mel- (1) "soft." Also in Middle English "dissolve" (of salt, sugar, etc.), "corrode" (of iron), "putrefy" (of flesh). Meaning "pass imperceptibly from one thing into another" is by 1781. Related: Melted; melting.
Figurative use "to diminish, wane; be touched, grow tender" is by c. 1200; transitive sense of "soften" (to love, pity, tenderness) is by early 14c. Of food, to melt in (one's) mouth is from 1690s. Melting point "degree of temperature at which a solid body melts" is by 1807. Melting pot is from early 15c.; figurative use from 1855; popularized with reference to immigrant assimilation in the United States by the play "The Melting Pot" by Israel Zangwill (1908):
DAVID Yes, East and West, and North and South, the palm and the pine, the pole and the equator, the crescent and the cross—how the great Alchemist melts and fuses them with his purging flame! Here shall they all unite to build the Republic of Man and the Kingdom of God. Ah, Vera, What is the glory of Rome and Jerusalem where all nations and races come to worship and look back, compared with the glory of America where all races and nations come to labour and look forward!
1854, "molten metal, a substance in a melted condition," from melt (v.). In reference to a type of sandwich (typically tuna) topped by melted cheese, by 1956, American English.